Logarithmic Forms and Diophantine Geometry Book

Preface     ix
Transcendence origins     1
Liouville's theorem     1
The Hermite-Lindemann theorem     5
The Siegel-Shidlovsky theory     9
Siegel's lemma     13
Mahler's method     16
Riemann hypothesis over finite fields     20
Logarithmic forms     24
Hilbert's seventh problem     24
The Gelfond-Schneider theorem     25
The Schneider-Lang theorem     28
Baker's theorem     32
The [Delta]-functions     33
The auxiliary function     36
Extrapolation     39
State of the art     41
Diophantine problems     46
Class numbers     46
The unit equations     49
The Thue equation     52
Diophantine curves     54
Practical computations     57
Exponential equations     61
The abc-conjecture     66
Commutative algebraic groups     70
Introduction     70
Basic concepts in algebraic geometry     73
The groups G[superscript a] and G[superscript m]     74
The Lie algebra     76
Characters     78
Subgroup varieties     80
Geometry of Numbers     82
Multiplicity estimates     89
Hilbert functions in degree theory     89
Differential length     93
Algebraic degree theory     95
Calculation of the Jacobi rank     97
The Wustholz theory     101
Algebraic subgroups of the torus     106
The analytic subgroup theorem     109
Introduction     109
New applications     117
Transcendence properties of rational integrals     124
Algebraic groups and Lie groups     128
Lindemann's theorem for abelian varieties     131
Proof of the integral theorem     135
Extended multiplicity estimates     136
Proof of the analytic subgroup theorem     140
Effective constructions on group varieties     145
The quantitative theory     149
Introduction     149
Sharp estimates for logarithmic forms     150
Analogues for algebraic groups     154
Isogeny theorems     158
Discriminants, polarisations and Galois groups     162
The Mordell and Tate conjectures      165
Further aspects of Diophantine geometry     167
Introduction     167
The Schmidt subspace theorem     167
Faltings' product theorem     170
The Andre-Oort conjecture     171
Hypergeometric functions     173
The Manin-Mumford conjecture     176
References     178
Index     194